JOURNAL OF 'THE ENGINEERING MECHANICS · PDF fileJOURNAL OF 'THE ENGINEERING MECHANICS DIVISION LARGE-DEFLECTION SPATIAL BUCKLING OF ... studIes of large deflectIons and post-buckling

10247 DECEMBER 1973 EM6

JOURNAL OF 'THE ENGINEERING

MECHANICS DIVISION

LARGE-DEFLECTION SPATIAL BUCKLING OF

TIDN-WALLED BEAMS AND FRAMES a

By Zdenek P. Baiant, I M. ASCE and Mabjoub EI NJmebiz

INnIOOUC'TION

As is well known. members of most metallic structures, reinforced polymer structu~es, .and som~ concrete structures must be analyzed. in the simplest approxImatIon. as thID-walled beams whose cross section exhIbits significant out-of-plane warping due to torsion (28.38,40). Planar failure modes of such stru~tures are known .to be an exception rather than a rule. Yet, nearly all studIes of large deflectIons and post-buckling behavior have so far been confined to planar (or to local) deformation modes and the buckling studies of thin-walled structures have been restricted to the bifurcation problem, to small deflections. and (exce~t for a few ~pecial cases) to straight members. An exception are a few ~tudle~ (10,14,18) ID which the post-buckling behavior in lateral buckling of straIght thID-wall beams was investigated analytically. solving by the Galerkin method a nonlinear differential equation based on some simplifying assumptions (10). However. the analytical approach would be too complicated in the case of complex structures.

The ~im of the pres~nt study is to develop a general matrix stiffness analysis of spatIal large deflectIons and of post-buckling behavior of thin-walled straight or curved beams, frames, and arches that are not excessively slender. This ~ev~lopm~nt will, of course, also enable analysis of spatial (lateral) buckling ID bIfurcatIon problems for arbitrary, not too slender. curved beams and arches, for which no method is presently available. The formulation will be given for

N<?te.-Discussion open ~ntil ~ay I, 197~. To extend the closing date one month, a wnl!en request must ~ flied with the EdItor of Technical Publications, ASCE. This paper IS part. of the c?pynght~ Journal of the Engineering Mechanics Division. Proceedings of the Am~ncan Socle~y of CIVil Engineers, Vol. 99, No. EM6. December, 1973. Manuscript was submitted for revIew for possible publication on December 28, 1972, and was revised on August 8, 1973.

',:ese~ted at t~e October 16-22, 1972, ASCE Annual and National Environmental Engmeerlng Meetmg, held at Houston, Tex.

: Prof. of Civ. Engrg., Northwestern Univ., Evanston, Ill. -Grad. Student, Dept. of Civ. Engrg., Northwestern Univ., Evanston, Ill.

1259


an arbitrary asymmetric open cross section. The analysis of thin-waIled structural systems was first approached using

the method of transfer matrices (40) and the force (flexibility) method (7,11,12, 17,39,40). These methods are, however, much less versatile than the displacement (stiffness) method. The stiffness matrix of a thin-walled beam element seems to have been first presented in 1967 by Krahula (22), for the case without initial stress, and by Renton (34), who included the effect of initial axial force on bending stiffness. With inclusion of initial bending moments and initial axial force, it was derived by Krajcinovic (23), Barsoum and Gallagher (6), Powell and Klingner (31), Mei (26), and Rajasekaran (33). The stiffness matrix which includes initial bimoment will be derived herein. Apparently. it has not yet been given in the literature. probably because this matrix is needed only for the large deflection analysis. and also because no bifurcation-type buckling can be induced by a bimoment. Argyris and Radaj (2) deduced the stiffness matrix with account of secondary shear strains and elastic stiffeners. but without initial stresses. The mass matrix was given by Krajcinovic (23) and Mei (26).

While most authors assumed cubic variation of all displacement parameters along the member. Krajcinovic (23) argued that exact static influence functions must be used instead. claiming that cubic polynomials lead to large errors whenever the ratio of Saint-Venant torsional rigidity to warping torsional rigidity is not negligible. This claim. however. has been disprOVed by Mei (27), as well as by the numerical results of the writers.

The relation between the warping parameters and bimoments at the coincident ends of two elements meeting at an angle apparently has not yet been determined and will be derived herein. It is needed for the analysis of curved members approximated by a series of straight elements. and also for the incremental loading approach to large deflections of beams which are straight prior to loading. Because it will be convenient to refer some element displacements and forces to the shear center and others to the centroid. a transformation matrix between these two frames of reference will have to be deduced.

It is well known that sufficiently lona shells can be analyzed as thin-walled bars. In the case of open cross section, which will solely be considered in the sequel, the analysis is reduced to a one-dimensional beam problem with the help of three simplifying assumptions:-(1) The cross section is assumed to be perfectly rigid in its plane while free to warp out of its plane; (2) the shear strains in the middle surface of the shell are neglected (Wagner's assumption); and (3) the transverse normal stresses and all bending moments in the shell wall are neglected. Furthermore, the material is assumed to be linearly elastic and the strains small, although the deflections and rotations may be large.

The first two assumptions yield for the longitudinal displacement. uz• at any point of the cross section (38,40)

U z = t(z) - ~' (z) x(s) - 'ii'(z) y(s) - 9'(z) (;)(s) ............. (1)

in which z = the length coordinate of the beam (Fig. I); primes are used to denote derivatives with respect to z, e.g., 9' = d9/ dz; x. y = principal centroidaJ

EM6 LARGE-DEFLECTION SPATIAL BlJCIa.M\IG 1261

axes of the cross section forming a right-handed system together with z (Fig. 1) [in. contrast with the more ~o~n use of a Ieft-.llanded system (40)]; t = z-dlsplacement at the centrOid; 1;, Tj = x and y displacements at the shear center C (40); S == rotation of cross section about axis z; and Ii) = principal sectorial area coordinate for a pole in shear center C (38,40), defined as Ii)

= n p(s')ds' where p == distance of waD tanaent from C, and s = length ~oord~~te of cross section measured from the sectorial nuD point (40); pds' IS posItIve when ds' turns radius p counterclockwise las 9 in Fig. 1), which is counter the usual sign convention (40). AlI quantities rdaTed to the shear center rather than the centroid are labeled by bars, and Ullillbeled quantities refer to the centroid. Quantities of both types will be mixed because this will lead to simpler final expressions. (Note that Eq. 1 is equivaleal to the usual expression , - E' x - TJ' y - 9' w in which aD quantities refer to 1M centroid.) Coordinate Ii) differs only by a constant from the warping functioa, w., of Timoshenlw and Ocre (38). Their use of b-.s is different. The JIIeIICIlt notation, which is consistent with refs. 7, 8, 21, 23, 40, is more concise in case of asymmetric cross sections.

P,C

FIG. ,.~ ~ _ .... ....................... Forcaa In~'" .....

zJ-: FIG. 2.--0.. .... _Iulln ... 01 ~WIIIIed ..... s.nent ce' Nat Shown)

MathematicaUy, Eq. I can be regarded as an assUlllption for the first four terms of Kantorovich direct variational method, with t, x(s), y(s), Ii)($) as the chosen orthOl0nal functions (8).

Consider now that the beam is initially in equilibrium at normal stress (10

due to initial axial force po (positive for tension), initial bending moments M O ~ about principal centroidal axes x and y (of sign shown in Fia. 1), and itriuai bimoment BO referred to the shear center. Thus (38,40)

po MO W flO (1~ = - + _x Y - -y x - -w

A I. I, I .. ....................... (2)

. . -0' 10 which B == - f A (1~ w dA; A = area of cross section; I. = f y2 dA and 1 y == I A x2 dA are the moments of inertia about centroidal axet x and !;.and .1 .. == J A 1i)

2dA = principal sectorial moment of inertia. (Quantity I.

IS IdentIcal With the warping constant, C .. , of Timoshenko and Ocre (38).] Subsequently. smaJI increments of loads ~ superimposed upon the loads


in the initial state, producing displacements ~, Tj, ,. 9, now considered as displacement increments with regard to the initial equilibrium state. The associated incremental potential energy is

~u= ~u, + ~uo - ~w ........................... (3)

in which ~ W = work of load increments on 1;, ... , 9, and

au,=f f ~Ee~dAdz+f ":'GI,9'2dz ................ (4) z A 2 z 2

~ U 0 = f z f A (1°zE , dAdz + f z M~9' dz . . . . . . . . . . . . . . . . . . . (5)

in which ~ U, = incremental strain energy; ~ Uo = energy due to initial stresses; GI. == Saint-Venant's rigidity in simple torsion (38,40); M~ = initial Saint-Venant's torque; t z = small (linearized) strain; and E, == finite strain, i.e.

au, I (au.)2 I (aU y )2 t,==~; E,=t'+2" az +2" ~ .............. (6)

in which u., u y = displacements of a general point of cross section in";rand y-directions. According to the assumption of rigid cross sections

u. = ~(z) - (y - Yo)9(z); u y = Ttz) + (x - xo) S(z) ........... C

in which xo' Yo = coordinates of shear center (40). Eq. 4 involves small strain ez because only infinitesimally small incremental

deformation is considered. Nevertheless, in Eq. 5, the finite strain expressior must be used, because, in view of (1~ being finite (large), (1~ e, would be accurate only up to small quantities of the first order in the displacement gradients while Be; is a small quantity of second order (9). But component 1 /2(au z/ az)2 oj E z has been deleted from E z since for small incremental strains it is negligible with regard to t z' The work of initial Saint-Venant torque M~ in Eq. 5 include! no second-order small term .

Substitution of Eq. 7 into Eq. 6, and Eqs. 2 and 6 into Eqs. 4 and 5, lead! to the foUowing expressions

f I -~U, = 2" [E(A,'z + lyl;nz + I.i1 n2 + I.,9 n2

) + GI.S'2]dz ...... (8) ,

in which

EM6 LARGE-DEFLECTION SPATIAL BUCKLING 1263

13 = _I_J Y(X2 + y2)dA - y; ~ = _I_J x(r + y2)dA - x (10) y 21 (J , 21 0

• A Y A

Note that for a doubly symmetric cross section W = O. and so ao does not contribute to ~ U o'

Because the preceding equations for incremental potential energy in the presence of initial stress are not available in the literature. it has been checked that the Euler differential equations obtained from Eqs. 8 and 9 by variational calculus are the same as those derived by Vlasov from considerations of adjacent equilibrium configurations (Eqs. 1.10 of Chap. V or 1.28 of Chap. VI, and Eq. 15.3 of Chap. V, in Ref. 40). One can also verify that the natural boundary conditions associated with Eqs. 8 and 9 are the same as those resulting from Eq. 16. Note that Eqs. 9 and 10 apply for an arbitrary nonsymmetric cross section.

In the practically prevalent case of a cross section symmetric with respect to axis y, the foregoing equations simplify since Xo = ~, = 0; w = w + Yox; Yo = - 1",,1 I" in which I ... = JAW x dA, and w = sectorial area coordinate

referred to the centroid; a = B + M, Yo' in whi<:b B = - I AUt W dA EI 9" = bimoment referred to the centroid; and t = ~ - 9 Yo' il = TJ. In the"'case of double symmetry. Yo = 13, = o. w = w. a = B. and ~ = ~.

The stiffness matrix will have by far the simplest form if some quantities are referred to the shear center and others to the centroid. as has been alread~ introduced (in contrast with the current practice in finite elements for beams) Referring to Eq. I. it is clear that the displacements in the cross section arl fully characterized by seven generalized displacements ,. ~, il. 9. - il' . ~' 9'. of which the first six describe the rigid body motions of cross section and the last one is the warping parameter. Thus, the column matrix, q. 0

generalized displacements of a beam element (Fig. 2) has the components

q=('i't,ili.9i,-il/,~/.6/. Cj'~j."lj.ej.-il;.~;.9i>T ...... (II in which subscripts i. j refer to end cross sections i. j and superscript T stands for the transpose of the matrix.

For approximation of the stiffness matrix of the element. C. ~. il. and e will be assumed as:

C=a,+b,z; ~=a2+bzz+c2Z2+d2ZJ , ........ (12)

Tj=a)+b Jz+c)z2+d)zJ; 9=a4+b4z+c4Z2+d4ZJ

in which ai' b l' a2 , ... , d 4 are arbitrary constants. Altogether. there are 14 arbitrary constants. which can thus be expressed in terms of the 14 components of q. giving a relation of the form

(,.Lil.9)T= N(z)q ............................ (13)

in which N(z) = (4 x 14) matrix whose coefficients are certain cubic polynomials in z (matrix of shape functions). To ensure convergence. the rigid body rotations without straining and the states of constant ~". il". " • e' . Ow must be available.


It is easy to check that Eqs. 12 satisfy these conditions. . Substituting Eq. 13 into Eqs. 8 and 9, incremental potential energy 6 U (Eq.

3). is obtained as a quadratic form in q. As is well known. ~he .elements of incremental (tangential) stiffness matrix K = [KijL (14 x 14) In size. are then

obtained as

a2 b. U KII = -- ................................. (14)

aqjaql

Carrying out the differentiation, the incremental stiffness matrix is obtained as a linear function of the initial internal forces. i.e.

K - K + noK + MOK + MOK + aOK •................ (15) - 0 r~ I x 1 y 3

in which Ko depends on the elastic moduli and is called elasti~ stiffnes.s matrix, while matrices K I' K 2' K 3' K. are independent of th~ elastic modulI ~nd are caUed geometric stiffness matrices. All of these matnces are symmetric. The nonzero elements of the upper triangular part of K are

EA 12k, 6 0.

K - K - K ---' K = -K29= K 99 =--+-P, 11-- 18- 88- I' 22 ,] 51

6 ° 6y o o. K K 2 4 == - K 2 II == K 9 II = - K 49 = - SI M. + 51 p, K 26 = 2 13 =

6k, po K -Ku= -K9iJ =r+lo; K 2 ,= K 21.= -K91.= -K611 = 46

1 YoPO '"" K. 13 == - K 79 = - K II 13 = -10 ~, + 10; K) ) = - K) 10 = K 10 10

6 K14 = -K11I = -K. 10 = K 1011 = - 51 MOy;

H, po K) s =' K 3 12 == - K s 10 == - K 1012 = - r -10

= 12+-K 2 - +--PO+--Mx--WB, ( (i) C J 6r 12~, 0 6 - -0.

5 I) 51 51 51

-1 K.s= K. ,2 = -KII12= -Ksll = -K 71O =1o ~Y;

4k 2 K = K = __ x + - lPO;

55 1212 I 15

EM6 LARGE-DEFLECTION SPATIAL BUCKUNG

-2 1 Ks,=KI214=-M~; K s1 .. =K,12=-Mo;

15 , 30'

2 k" I 4k 21 KSI2 =-- --PO; K66 = KI3I3 =--' +-po.

1 30 15

2 2 K 6,= K 1314 = --~+-YopO;

15 15

1 I K614 = K'I3 = 30 ~ - 30 YoPO;

in which I = length of element

2k, I K =---po.

613 I 30 '

_ W C=GI.; C,=EI .. ; W=

I ..

1265

(17)

Note that K'I = K ,_,.1_7 or K ,_7.1+, for i> 7. Also note that Eq. 16 applies for arbitrarily asymmetric cross section. It has been checked that the special case o! stiffness matrix for a planar case and a mono-symmetric cross section with BO = 0, and for a simplified account of simple torsion, as presented in Refs. 6 or 31, agrees with Eq. 16. The special case of stiffness matrix without initial stress, presented in Ref. 26, is also in agreement.

The column matrix, F, of associated incremental generalized forces at the ends of the element ensues from the condition that F T q must give the correct value of work for any q. This is true if, and only if

F= P V V - - - - - - T ( i' Xi' Y/' Ml;' M x ,. M", B,. PI' V XI ' V yJ ' M ,j , M x, ' M yj • B j) (18)

in which P= incremental axial force (acting in the centroid); V , V = incremental shear forces (acting in the shear center); Iii = incremental tor~e about shear center; My. M'L = incremental bending mo~nts about centroidaI axes x and y (Fig. 2); and B = - fA (J, w dA = incremental principal bimoment, referred to the shear center.

The simplest method of analysis of a curved beam is based on approximating the beam by a series of straight elements. For this purpose it is necessary to deduce the transformation matrix, T, which transforms the generalized forces and displacements at the end j of element ij to those at the end j of the adjacent

1266 DECEMBER 1973 EMS

element jk meeting at an angle at the joint j (Fig. 3) or, alternatively, to global coordinates. All elements of matrix T can be determined from kinematics, except those which mUltiply 6' or ii. The latter ones seem to have eluded attention so far. They cannot be ascertained merely by consideration of u, -distributions at the end j of elements ij and jk (Fig. 3) because these distributions are incompatible for any nonzero angle between the elements. This fact need not be viewed as objectionable because the finite elements approximate, in the first place, the one-dimensional curved beam problem, and not the actual three-dimensional problem which does not involve the simplifying assumptions

of thin-walled beam theory. In view of this fact, it will be imagined that the approximation by straight

elements is the limiting case of a beam consisting of straight elements interconnected by means of short curved elements (Fig. 4) whose length L tends to

AG. 3.-Inc:ompatibiIity of Assumed Warping Oispa.c.ment DIstributions ". when Two Elements MeM lit Angle

a) b)

FIG. 4.-Approximlltion of Curved BMm (e) by Beam Consisting of Stre~ht Bementa (b). end Trensttion from (el to (bl Through e Beam with Streight and Curved Segments (el

zero. To this end it is necessary to find the transfer matrix, T' , of the curved element, jf, in Fig. 4, defined by the relation

Sj,=T'Sj .................................. (19)

in which subscripts i, j' refer to cross sections j, j' (Fig. 4); S = the state vector for the cross section, which may be defined in the case of circularly

curved members (40) as

S = ( t. t. Ti. 6. - Ti ' , ~' • 6' + ~. p. V x' V y' M" M x' My. B) T • • • (20)

For examining the transformation of iJ and 6' , the connecting element, jf, will be considered as a circularly curved thin-walled beam of radius R, with the center of curvature on the positive side of axis x and cross section symmetric about axis y. No initial forces and no distributed loads will be considered. Deformations of this beam are known to be governed by the differential equations (Eq. 1.10 of Chap. XII in Ref. 40):


........ (21)

in which c2 = L 2 EI xl EI.,; K 2 = L 2 G lsi EI.,; and 9 = rotation about shear

center. The exact transformation matrix, T'. has been derived by integration of these equations and is given in the Appendix. Letting L -+ 0 and R -+

o while keeping <1>; = LI R constant (Fig. 4). the limit of T' is the sought for transformation matrix. T.

In the manner just explained it has been found that

T = lim T' =

R: ------------

oR: ------------

o I o

----------

o Ro

o

I. I I

- - - - - - - - t- - - - - - -. - -R- - .- - -o • 0 ------------ ----------

o I

......... (22)

in which R = the (3 x 3) rotation matrix as given in Ref. 32. As is seen, T has a very simple form, for which T -I = T T. The same result can reasonably be expected to hold for a noncircular nonplanar curvature and in presence of initial stress. Aside from the columns and rows pertaining to 9' and B. the structure of T can. naturally, be determined from rigid body kinematics.

The unity coefficients in Eq. 22 can also be obtained by simple intuitive reasoning. If the columns relating to 9' and B had nonzero elements outside the diagonal, warping 9' would produce a rigid body displacement in the same joint and bimoment B would result in an internal force or moment in the same joint. both of which would be unacceptable features. So only the diagonal terms of the columns considered may be nonzero. If the rows relatiJl8 to 9' and B had nonzero elements outside the diagonal, rigid body displacements could not occur without causing a warping in the same joint. and internal forces and moments would cause a bimoment, both of which are again impossible. Thus only the diagonal terms of the considered rows may be nonzero. Consider now that 9; == a 9;. The same transfer coefficient must apply in the reverse direction. i.e., 9: = a 9;. This is possi.!'le only if a = l. Similar considerations show that the diagonal coefficient for B must be also I.

The foregoing argument, however. does not verify consistency with the differential equations of the problem, which is an important aspect to be checked because the u, -distributIOns in the cross sections of the interacting straight elements are incompatible. Alternatively, Eq. 22 could also ~ derived from the natural boundary conditions of curved beam segment ij (Fig. 4). Note that Eq. 22 has already been implied in Ref. 21 without offering any justification.

The simple form of matrix T makes the approximation of curved beam:; by straight elements particularly advantageous.

Beams of variable cross section may be approximately treated using elements of constant cross section which represent the average cross section of the actual


b 'thi the I'nterval considered. The u -displacements from the adjacent eam WI n t. .th th d' I t e then Of course incompatible. But it is consistent WI e or mary e emen s ar" "d that

differential equations for beams of variable cross se~tlon (7) to consl ~r 9' and B are continuous between the elements of different cross section, as has been proved in Ref. 7.

TIWa'OIIMA11OIITO Gt.oua. eoo.....TES AND TO CINTRoIDAL AxEs

Rather than transforming q from local coordinates x •. y •. z of one ele~ent into the local coordinates x, y, z of the adjacent element, It IS more conv~ruent in practical computations to transform q of each element ~nto glob~ coord mates X, Y, Z of the structure. Observing that the transformatIOn matnx for e~ement displacement column vector q is the same as for state vector S, one obtams

• T .............. (23) q = q .................... .

. h' h • - column matrix of element displacements transformed to global m w IC q - .th f trix R coordinates; and T = matrix of form of Eq. 22 WI rota Ion ma

R =r:: :: ::] ............................. (24)

L~x cy Cz

b b b) ( ) are the global coordinates of in which (ax,ay,az)' ( x' yo z' cX ' cy , Cz the unit vectors a, b. e of the axes x. '1, .t of the elem~nt.

The transformation matrix. T, given by Eq. 22 apphes, however, ~nly w~en all components of q are referred to the same point in the c~oss .se~tlon. which . nl for doubly symmetric cross sections. Otherwise, It IS necessary ~~ :=s:or! first displacement matrix q into matrix q C all of ~hose compon~nts are referred to the cross-sectional centroid. This transformation may be wntten as

qC=tq ..................... . .....•.....•.. (25)

in which t = (14 x 14) matrix all of whose diagonal terms = ~ and, taking a monosymmetric cross section as an example, ~e nonzero off~l~onal tferms

. - and ~, - t' + 9' Y Similarly orce represent the relations E = E + Yo' '" - "'. . o' '. matrix F is transformed into F C and, in conformity With the general rules m matrix structural analysis

F = t T F C • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (26)

That this is indeed so may be checked by deriving t T directly from ~he fa~t that nonzero 0 -...-the ff -A;Oftonal terms of t T must correspond to the relatIons M z

- M + V y B- = B + M y in the case of monosymmetry. - 0' y o· . th f fon Th~s, fo; asymmetric or monosymmetric cross sectlon~. e trans orma I matrix to global coordinates which also includes tranformatlon of all components to the cross section centroid is

....... (27)

fLoe matnx' .nven by Eq. 22. For example, in the case of a in which T 1 = UI 0"

EM6 LARGE-OEFLECTION SPATIAL BUCKUNG 1269

monosymmetric cross section, T can be obtained by carrying out the matrix multiplications in Eq. 27. Th!s yields

R

T=

-yoa y

yo(az - by) yoa, yoa.

, -yoe, 1 ______________________ _

, , , R -yob.

YoO - e.) ----------------------- -------"

I

(28)

ditto

The stiffness matrix and the column matrix of generalized forces are then transformed to (or from) the sIobaJ coordinates as

K* = T-1TKT- 1; Fe = TTF*c ...................... (29)

The second relation foDows from the identities A U = q TF = q * TF * = q TT TF * , valid for any q, and the first one foDows from the identities F = K q = T TF * = T T(K *q*) = TTK * T q, also valid for any q.

Matrices K * of the individual elements in global coordinates are finaUy assembled into the structural stiffness matrix, i, by the standard technique (41). The incremental equations of equibbrium of the structure read it q = F, in which q assembled column matrix of all generalized displacement increments of the structure; and F = the associated column matrix of given increments of applied generalized forces.

INa. TElllTAL l.OADNa ~_IN T ... o..r.o.

For spatial large deflection analysis the same methods as for planar problems (1,19,24,29,37,41) can be applied. As is well k.nown, two different methods of incremental analysis exist. In the first method the incremental properties of the material and the elements are all referred to the natural unstressed state [Lagrangian coordinate approach (15)] while in the second method they are referred to the current stressed state [Eulerian coordinate approach (15)] or, approximately, to the stressed state at the beginning of the current small time step, as is done herein for the purpose of numerical analysiS. The first method is suitable, as a rule, when strains are large. But in the case that only displacements and rotations are wac while material strains always remain small, as is assumed in this study and holds for nearly all practical thin-walled structures, the second method may be more expedient. In contrast with the first method (29,41), it has the ad~antage that the incremental stiffnesses depend onJy upon the current stresses (or, approximately, upon the initial stresses <To of the current small increment, as indicated by Eq. 15) and are independent of the current displacements. This saves deriving several more types of matrices, 14 x 14 in size, which more than outweighs a disadvantage consisting in the fact that the second method requires the local coordinate systems of the elements to be up-dated

1270 DECEMBER 1973 EMS

at each small load increment. Therefore, the second method has been adopted in the present study.

The loading vector, f, is gradually incremented in small increments A f (r) = f

U+

1J - fIr) (r = 1, 2, ... ). The state of the stru~e at the beginning of

the rth increment is characterized by global coof(l~~tes X j ,,).' Y j,!), Ziln of aU joints i = I, 2, ... N

" by unit vectors bn(r) 8lV1ng the Ulrectlon of the

y-axis of the cross section of the nth element (n = 1, 2, ... N 2)' and by the iru·t.·aI internal forces P , M , M , H(r) in each element (Fig. 5). The ( r) X(r) Ycrl . • computation of the changes due to the rtn increment may be arranged In the following steps:

I Us.·ng no = P ~ = M ~ = M ,8° = H(,), stiffness matrices . r (r)' ~ X(r)')' Y(rJ.

K givenbyEqs.15and 16 are computed for au elements (n = I, ... N 2 )·

2. Unit axial vector en", of each element is determined from joint c?Ordinat~s X Y Z and using vector b and a = c x b , rotatlOn matnx

j( r, , J(r) , j( r) , 11(,) II( ,) lIt Q "< r) •

R , Eq. 24, is set up for each element. The length of element may be determmed fr~~ X y. Z., but approximately may usually be taken the same as

Jt r) , J~ r) , J( r)

in the natural state. 3. Stiffness matrices K· in global coordinates are computed from Eq. 29

and are assembled into the structural stiffness matrix, taking boundary conditions into account. Displacement increments qir) due to load increment Af(r) are then solved from a system of linear equations and the coordinates X Y Z. of joints at the end of increment are evaluated.

)\,. i.I' ),,+ I)' 1,,+ I) ed F * 4. Internal force increments at the ends of elements are comput as

= K * q* for all elements and are transformed into local coordinates accordina to Eq. 2~/ Then the average increments of p, M z' My, and B within the elements, and the final average values of p(r+Il' M Z\'+')' My,,,, within the element, are determined for all elements (as averages between the v~ues at ends of elements).

5. Unit vector b of the y-axis of each element at the end of increment may be determined''ir~m the average incremental rotation 8 n = (8 1 + 8)/2 within the element about its axis. This rotation transforms vector b"(.) into the vector

b~ = bill') + (cn\r) x b"w) an .....•................•.. (30)

which. however, is usually not exactly perpendicular to the x-axis unit vector C determined from XI ' ... Z. . Nevertheless, vector b~ with c n '+')

"(r+I) (r+ I) J(rTI) • thi ~, determines plane (y x) at the end of increment and vector b n,,+ .J.!n s plane may thus be ascertained as the vector which is normal to C"\Hn· 1nus

b = (c x b' ) xc ..................... (31) "(rod) "(,+I) "("+') "(,+\)

Note that rotations ~' and - TI '. if viewed as infinitesimally small, have no effect on orientation of plane Y1:.

6. Due to the numerical error arising from the finite size of the incre~ents, there is. always a certain small imbalance of the total forces (determined directly from to~d.isplacements) acting at each joint at the' end of increment. To remove this error, the I alance force must be subtracted from the given load increments in the subsequent i tion of this step.

7. In the foregoing proceOu..~ all element properties are based on the state at the beginning of the rth increment, at1d so the computed increments are


accurate only to the first order, in ~f, similarly as in the Euler method for ordinary differential equations. To reduce the error to the second order, a procedure similar to the second-order Runge-Kutta method may be used, i.e., steps I to 6 may be iterated basing all element properties on the state in the middle of the increment as determined from the previous analysis of the rth increment. Thus, denoting the averages between the" initial and the final value in therth step by subscript (r + 1/2), e.g., 8(° 1/2) = (8 (B )/2 ) _ _ r+. (r). (r+ I) ,

the values pi = P,,+ 1/2)' ... , J30 = Bu+ 1/2) are used 10 step I; 10 step 2, C" I 2 and R" "2 are determined from XI" •... and b ; in step 5 Eq:+30'is modified ~s '''',2' ", .. '/2, ,

b~ = b n,,, + (C n ,,.'/2) x b n"'1i2) 9n •.••••••••.•••••••••• (32)

while Eq. 31 remains unchanged. Due to numerical errors vector b is . "1'+ I, 2)

TABLE 1.-Numerlc.l Results for Bifurcation-Type LM-.I Budding of Straight .... rna

Number of Slenderness w,cr by finite Error, as a elements Lid Exact W, elements percentage

(1 ) (2) (3) u (4) (5)

4 10 627.40 627.89 0.08 8 38 101.4082 101.4106 0.002

16 38 101.4082 101.4089 0.0007 32 38 101.4082 101.4087 0.0005

Note: Moments are given in kip-in. (I J3 N-m).

AG. 5.-Straight-Sement Approximations at Various Stages of ~ Lateral Budding of I-beam (Numerical Example in Fig. 7,

not exactly of. unit length and must therefore be normalized before evaluating Eq.32.

8. Itellltions of steps 1 to 6 as outlined in item 7 are carried out until the difference between the results of two consecutive iterations becomes negligIbly small. Then the analysis proceeds to the (r + l)st increment. Usually it is not advantageous to use more than about three iterations since further iterations improve accuracy less than a reduction in the load increment.


TABlE 2.-4IIumerical Results for BIfurcation

..,. in Slenderness M:cr for Exact Case degrees Lid EI .. R_oo MO

(1) (2) (3) (4) (5) '" (6) a 30 12 4C I ~3,99S.2 4,734.6

-3,353.2 b IS 24 C I ~627.4 748.6

-S24.1 c IS 12 4C I ±3,99S.2 4,387.1

-3,696.4 d 7.5 12 4C I ~3,99S.2 4,165.6

-3,823.3 e 7.S 12 C I ~2,128.S 2,156.3

-2,000.8

Note'"' Moments are liven In kip-m. (113 N-m).

y

AG. 8.-SlmpIy Supported Beem or CIrcut.r Arch Under Uniform I~ Bending Moment

1be foregoing procedure is applicable only on the increasing branch or on the. decreasing branch of the load-displacement curve. At the limit point, at which the structural stiffness matrix becomes singular, it is possible to use various weD-known techniques described, e.g., in Ref. 16.

1be method presented in this study can be also applied to problems with inelastic strains (due to temperature, plastic yield, or creep), if the inelastic stresses are converted to applied loads.

1be accuracy of the proposed method of analysis has been examined by comparisons with exact analytical solutions. Such solutions exist only for bifurcation-type buckling problems and only for a few simple structural shapes and loadings.

BifumItioo-Type BencIing-Torsional Budding and Lateral Budding of StraIght Columns and Beams.-First the case of bending-torsional buckling of a straight pin-ended column under uniform axiaJ force pO has been considered. 1be cross section properties were El- = 1,024 x 106 kips-in 4 (l 896 kN_m 4

) 01 = 60,000 kips-in 2 (172,190 N_~2), Ely = 2,580,000 kips-in'2 (7,404 kN~m2): Xo


Type Lateral BuddIng of CIrcu .. , Arches

Ec' as a M~a by percentage Error, 8S 8

finite elements (Eq.35) percentage Error/E c (7) (8) (9) (10)

5,354.8 18.5 \3.\ 0.7\ - 3,068.2 -16.0 8.3 0.52

879.9 19.3 17.5 0.908 -454.7 -16.4 -13.2 0.805 4.467.3 9.81 1.83 0.187

- 3,616.9 -7.48 -2.15 0.290 4.215.7 4.34 0.90 0.208

- 3,793.2 -4.34 -0.68 0.157 2,352.3 6.1 - 4.2 0.689

-1,929.8 -6.1 -3.1 0.509

= Yo == O. Dividing the column of length 640 in. into eight elements, the computed bending-torsional buckling load differed from the exact value given by the formula in Ref. 38 only by 0.02% (upper bound result). For the two elements and a length of 160 in. (4,064 m) the error was 0.62%.

Then the cases of lateral bucklins of beams were investigated. Simple exact formulas exist oruy when the beam is simply supported and ~ is constant along the beam, i.e., the beam is loaded by couples at its ends. The cross-sectional properties were the same as before. The error of the buckling load, PO, with regard to the exact solution accordins to the formula given in Ref. 38 is tabulated in Table 1 for different numbers of elements. (Lj d = span to beam depth ratio.) Obviously, excellent accuracy is easily attainable in the case of straight beams.

The eigenvalue problems arising in the analysis were solved by standard library subroutines. But when the size of the matrix was too large, the dependence of a certain suitably chosen stiffness parameter on the initial load was determined instead and its zero was found by the regula falsi method.

Bifun:atIoD. Type LatenI Buck", ~ ..\I'da.-Some simplified practical problems for lateral stability of arches have been solved numerically (30,36) for the case of simple torsion. Among thin-walled curved beam problems, a simple analytical solution exists oruy for a circular two-hinse arch, curved in plane y z and subjected to uniform initial moment ~ introduced at the ends which are free to rotate in plane y z but prevented to rotate in other planes (Fig. 6) (40,20). According to Chapter XII in Ref. 40, the smallest positive or negative critical moment is

~a = a ± Va 2 + ~ 2 . . • • . • • • . • • • • • • • • • . • • . • • . • • • (33)

in which the plus sign is for bending in the sense of initial curvature; the minus sign is for bending against it; and

a =_1_ (EI + C+~CI)' 2R ' L2

1274 DECEMBER 1973 EM!

13= (~ __ I )EI C(I +1T2L2~) .................. (34)~ L2 R2' C

1

L == length of the arc between supports. The cross section properties, GI, and EI" considered in the practical examples,

were again GI, = 60,000 kips-in 2 (172,190 N-m 2), EI, = 2,580,000 kips-in 2

(7,404 kN-m 2), Xo = Yo = 0, while two values of EI", were used as shown in Table 2 with C

1 = 1,024 X 10 6 kips-in4 (1,895,900 N_m 4

). It has been found that with increasing numbers of elements the numerical results for the critical M~ converge quite rapidly (approximately as in Table I), but frequently to a value which is substantially different from the exact solution. Some typical values of MO to which the finite element analysis converged (values for 16 elements) are'~~mmarized in Table 2, for various lengths L of the beam (measured along the arc), central angles 'Y of the are, and warping rigidities EI ...

In evaluating the numerical error, comparisons of the change in critical moment due to curvature of the beam were made. This change was characterized by the parameter

E = c

MO -W xC, %:)IC .................. (35)

in which MO exact critical bending moment for straight beam (R -+ 00) of the same c~~ss section properties and length; and ~ = exact critical bending moment of the given curved beam. The numerical an~ysis may be considered to be a useful tool when the ratio of the numerical error to change E c due to curvature, as defined by Eq. 35, is small; i.e., less than about 1/3.

In general, it is seen from Table 2 that the numerical error is unacceptably large when: (I) The curvature is large; (2) the arch is slender, i.e., the length-to-depth ratio is large; and (3) the warping rigidity is snWlI. In other cases (cases c and d in Table 2), the analysis proposed herein gives acceptable results. Below the buckling range the limitations are less stringent.

The fact that the numerical solution of buckling loads is poor for large curvatures and for slender arches is not surprising. Similar conclusions have recently been reached in the studies of finite elements for thin shells and for plane deformations of slender arches (3,4,5,13,25,35,37). From these studies it became clear that, in order to obtain a generally applicable planar finite element, a curved element will have to be developed. This need is probablY due to the fact that only curved elements can guarantee full continuity of finite element subdivision (while for straight elements the end cross sections forming an angle overlap on one side of beam axis and leave a gap on the other side). For convergence to a correct value the shape functions must satisfy exactly the conditions that: (1) no self-straining occurs at rigid body rotations; and (2) all states of constant generalized strains are available. Satisfying these conditions with a curved element exactly is not easy.

Nevertheless, for many practical problems the present formulation appears to be satisfactory. While arches rarely have deep members and smaIl enough curvatures, beams and frames do. They usually cannot develop very large curvature prior to failure of the material. In such cases the present large deflection analysis is applicable.


Large Deflection Lateral Buckling.-The foregoing method was applied to analyze post-<:riticallateral buckling of a simply supported I-beam under uniform vertical load (Figs. 5 and 7). The ends of beam were rotating freely in planes YZ and XZ, and translating freely in the Z direction, all other motions being prevented. The properties of the beam were: span L = 640 in., Ely = 2.580,000 kips-in 2 (7,404 kN-m 2

), GI. = 60,000 kips-in 2 (172,190 N-m 2), EI .. = 1,024

x tot' kips-in4 (1,896 kN-m 4), Elx = 234 x 106 kips-in 2 (672 x 106 N-m 2

),

Xo = Yo = 0, depth d = 40 in. (1.02 m). First, the critical load was found by computing the inverse rotation. 9 1 , due to a fixed lateral loading moment applied on a beam under various initial vertical loads; plotting these values

• Q.

.ao:

..J ... 117

0 20

C 0 -I

-I C ~ 0 10 ~

0 0

deflection increment

mz I. ,2mz / Ye /

I /5mz

I / I / I I I I II 1/ II

I

.2 .4

MIDSPAN ROTATION

.6 .8

8 1 (rod)

AG. 7.--Computed loIId-Rot8tion CuIV .. for targe.DIIIec:tIon Lat ... Budding ( .... suits for Three Different Verticlll Displacement Incrementa, and for Three DIfferent Fixed Lateral Disturbing Distributed Moments III. = 0.25 kips, 111./2. 111./5; 1 kip = 4,448 NI

gives curve a in Fig. 7 and the critical load is obtained as its intersection with the vertical axis by the regula falsi method.

For large-deflection analysis, the beam was considered to be disturbed by a small constant distributed applied moment m, about global axis Z, initially coinciding with beam axis. The plot of lateral rotations versus vertical load is given in Fig. 7 for. three values of the disturbing load. The increments of vertical distributed load were adjusted as to give constant increments of midspan deflection, for which three values were used. The results obviously converge with diminishing load increment.

One interesting observation from Fig. 7 is the fact that after exceeding the


c~tica1. load the beam stif~ens, in a manner similar to columns or plates. The stlfferung occurs at a relatively large deflection, so that the post-critica1 reserve of this beam is not usable. F~he~re, large-deflection buckling of beams with asymmetric angle cross

sections (Fig. 8) was computed and compared with the experimental results of Engel .and ,Goodier (14), which have not yet been fitted by any theory, to the wnters knowledge. The comparison is exceUent, which validates the present method of analysis.

Larae Twist of Initially Bent Beams.-Fig. 9 shows a comparison for channel

• A

2.0

o 1.0

o C o .oJ

O.S

o COMPUTED TUTS OF ENGEL AND GOODIER

HOft'ZONTAL DEFLECTION lin)

2

VERTICAL DEFLECTION I in)

. 1UIaC' TOTS ._- IIUa< S ICI..IIT1ON - PREIPfT WTHCO

~ 0 ~ ~ ~ ~ 00 ~ ~

TWIST AT MIOS~ (de9B •• 1

AG. 9.--comp.rison of PreNnt SaIution with ExperImem.I o.ta and AMIyticaI Solution of Black (10); Channel is Supported as In fig. 7 and is PreIoaded by Constant Uniform LoH cau.uag Moment M, (111Hn. = 113 N-mm)

EMS LARGE-DEFLECTION SPATIAL BUCKLING 1277

sections with the tests and an approximate analytical solution of Black (10). [Since he has not reported My (Fig. 9), its value was deduced by fitting the data]. Integration through the limit point (peak of the curve in Fig. 9) was made possible by switching to prescribed displacement increments. A second, minimum limit point exists for twist angles exceeding the range in Fig. 9.

SUIoWARY AM) CoNc:wsIoNs

I. The general incremental stiffness matrix of straight thin-walled beam elements of generally asymmetric open cross section, subjected to initial axial force, initial bending moments, and initial bimoment, is presented.

2. The expression of incremental potential energy, from which this matrix was derived, was checked for giving the correct differential equations of the problem (40).

3. The transformation matrix relating the forces and displacements at the end cross sections of two elements meeting at an angle has been derived as the limiting case of a transfer matrix for a curved member. Noteworthy is the fact that warping parameter 9' and bimoment B transform with unity and are independent of all other generalized displacements and forces.

4. A relatively simple treatment of arbitrary asymmetric cross sections is made possible by referring some of the element displacements and forces to the centroid and others to the shear center. The transformation matrix to global coordinates then also includes transformation of all components to centroidal axes.

5. The incremental large~splacement analysis is formulated using the Eulerian coordinate approach with up-dating of the local coordinate systems at each load increment. The beams, curved either naturaIly or due to prior deformation, are imagined to be composed of straight elements meeting at angles.

6. Numerical comparisons with known exact solutions of some bifurcationtype stability problems have indicated that the accuracy of the proposed scheme is excellent for straight or almost straight beams. In the case that the beams are curved, either naturaIly or as a result of large deflections, the scheme is sufficiently accurate only if the curvature radius, the depth-to-span ratio, and the warping rigidity are sufficiently large. This fact is not due to the presence of warping and is not surprising since similar limitations have been found in the case of planar deformations of arches. A possible remedy is the development of appropriate curved elements.

7. Because of the aforementioned limitations, the method presented herein is usually applicable only to large deflections of straight beams, frames or slightly curved beams in which slenderness is not too high and excessive curvatures cannot arise prior to failure of the material. Below the buckling load range the slenderness and curvature limitations are less stringent.

8. The present method gives excellent agreement with known experimental data.

9. A numerical example showed that in lateral buckling of an I-beam there is a large postbuckling reserve which, however, is usable only if very large deflection is permitted.

10. A snap-through was found to occur in large twist of an initially bent channel beam. It is followed by a minimum limit point.

II. The method can also be used for classical small deflection problems of


curved thin-wall beam structures without initial stress.

Computer funds provided by Northwestern University are gratefully acknowledged.

AIPeax I.-TRANSFER MAlRX OF BEAM CuMD ... PLANE

The row of the transfer matrix which gives bimoment B j' at end j' in terms of state vector at end j (subscript j) is characterized by the following equation that has been obtained by integrating Eq. 21:

R 2 c2 L shK

K(L 2 - c2 R2)

) ___ R_L_3_s_h_K _ + R(2L~2_sinc;(R;») l V., . .............. (36) l K(U - c2R2) )

This relation has been checked for giving at R - QC the bimoment row of the well-known transfer matrix of a straight beam, i.e.:

H; = - (eLK -I shK)9; + (L K- 1 sh K) M zj + (ch K)B j ••••••••••• (37)

The limiting case R -+ 0 gives I in Eq. 22.

APPErax II.-Rs suas

I. Argyris, J. H .• "Continua and Discontinua," Proceedings, Confere~ on Matrix Methods in Structural Mechanics, Air Force Institute of Technology, Wnght Patterson A.F. Base. Ohio, 1965.

2. Argyris, J. H., and Radaj, D., "Steifigkeitsmatrizen dunnwandiger Stiibe und Stabsys-teme," Ingenieur-Arcruv, Vol. 40, 1971, pp. 198-210.. ..

3. Ashwell, D. G., and Sabir, A. B., "Umitations of Certam C:Urved ~lJute Elements when Applied to Arches," International Journal of Mechamcal Scumces, Vol. 13,

1971, pp. 133-139. ., A I' . 4. Ashwell, D. G .. Sabir, A. B .. and Roberts, T. M., "Further Studies In the pp lcat~on

of Curved Finite Elements to Circular Arches," International Journal 0/ Mechanical Sciences, Vol. 13, 1971, pp. 507-517.


5. Austin. W. J. "In-Plane Bending and Buckling of Arches." Journal of the Structural Division. ASCE. Vol. 97. No. ST5, Proc. Paper S130. May. 1971 (Disc., Vol. 98. No. STI. July. 1972. p. 1670). '

6. Barsoum, R .• and Gallagher. R. H .. "Finite Element Analysis of Torsional and Torsional-Flexural Stability Problems." International Journal for Numerical Methods

,. in Engineering, Vol. 2. No.3, 1970. pp. 335-352. 7. BaZant. Z. P. "Nonuniform Torsion of Thin-Walled Bars of Variable Cross Section,"

International Association for Bridge and Structural Engineering. Publications, Vol. 25. 1965. pp. 245-267 (see also Inien)TSke Stavby, Prague. Czechoslovakia. Vol. 15. 1967. pp. 222-22S).

8. BaZant. Z. P .. "Pieces longues a voiles epais et calcul des poutres a section ddormable.·· Annales des Ponts et Chausstes. No. III. 1969. pp. 115-169 (see also Sta~'ebnicky Casopis SAY. Vol. 15. 1967. pp. 541-555).

9. BaZant. Z. P .. "A Correlation Study of Formulations of Incremental Deformation and Stability of Continuous Bodies," Transactions. American Society of Mechanical Engineers. Journal of Applied Mechanics. Vol. 3S. 1971. pp. 919-92S.

10. Black, M.. "Non-linear Behaviour of Thin-walled Unsymmetrical Beam-sections Subjected to Bending and Torsion." Thin-walled Structures. A. H. Chilver, ed .. John Wiley and Sons, Inc., New York. N.Y., 1967. pp. 87-102.

II. Borisov, M. D., Torsional Analysis of Beam and Frame Systems of Thin- Walled Composite Bars (in Russian). Izd. Lit. po Stroit.. Leningrad. U.S.S.R., 1970.

12. Bychkov. D. V .• Structural Mechanics of Thin-Walled Bar Structures (in Russian). Gosstroyizdat. Moscow. 1962; see also Analysis of Beam and Framed Systems of Thin-Walled Elements (in Russian). Gosstroyizdat. Moscow. U.S.S.R .. 1948.

13. Dawe. D. J. "Finite Deflection Analysis of Arches." Internationallournal for Numeral Methods in Engineering, Vol. 3. 1971. pp. 529-552.

14. Engel. H. L.. and Goodier. J. N., "Measurements of Torsional Stiffness Changes and Instability due to Tension. Compression. and Bending," Journal of Applied Mechanics, Transactions, American Society of Mechanical Engineers. Vol. 20. 1953. pp. 553-560.

15. Fung. Y. c., Foundations of Solid Mechanics, Prentice Hall. Inc .. Englewood Cliffs. N.J .• 1965.

16. Gallagher. R. H .. "Finite-element Method of Limit Point Calculation," presented at the October. 1972. ASCE Annual and National Environmental Engineering Meeting, held at Houston, Tex., (Preprint 1823).

17. Gorbunov. B. N .. and Strel'bitskaya. A. I., Theory of Frames of Thin-Walled Bars (in Russian). Gostechizdat. Moscow, U.S.S.R.. 1948.

IS. Gregory, M .. "Elastic Torsion of Members of Thin Open Cross-section," Australian Journal of Applied Science, Vol. 12, No.2, June, 1961. p. 1974.

19. Hutchinson. J. W .• and Koiter, W. T., "Post-Buckling Theory," Applied Mechanics Reviews. Vol. 23. 1970. pp. 1353-1366.

20. Kollar, L. "Torsional Buckling of Thin-WaDed Curved Bars (Shell-Arches)," Acta Technica, Budapest, Hungary, Vol. 40, No. 3/4, 1962, pp. 337-353.

21. Kollbrunner, C. F .• Hajdin, N., and Krajcinovic, D., "Matrix Analysis of Thin-Walled Structures," Report No. 10. Institute for Engineering Research, Verlag Leemann, Ziirich, Switzerland, 1969.

22. Krahula, J. L.. "Analysis of Bent and Twisted Bars using the Finite Element Method," AIAA Journal, Vol. 5, 1967, pp. 1194-1197.

23. Krajcinovic, D .• "A Consistent Discrete Elements Technique for Thin-Walled Assemblages," International Journal of Solids and Structures, Vol. 5. 1969, pp. 639-662.

24. Mar~al, P. V .. "Effect of Initial Displacement on Problem of Large Deflection and Stability," Technical Report ARPA E5 4, Brown University, Providence. R.I.. 1967.

25. Megard, G., "Planar and Curved SheD Elements," Finite Element Methods in Stress Analysis, I. HollHld, and K. BeD, eds., Tapir-T.U., Trondheim, Norway, 1970, pp. 287-318.

26. Mei, C., "Coupled Vibrations of Thin-Walled Beams of Open Section using the Finite Element Method," International Journal of Mechanical Sciences, Vol. 12, 1970, pp. 883-891.

27. Mei, c., "Reply to Krajcinovic's Discussion," International Journal of Mechanical Sciences, Vol. 13, 1971, pp. 739-740.


28. Nowinski, J., '''Theory of Thin-Walled Bars," Applied Mechanics Reviews, Vol. 12, No.4, 1959; up-<!ated in Applied Mechanics Surveys, M. N. Abramson, et al., eds., Spartan Books, Washington, D.C., 1966, pp. 325-338.

29. Oden, T., Finite ElernnJts of Nonlinear Continua, John WIley and Sons, Inc., New York, N.Y., 1971.

30. Ostenfeld, A., "Seitensteifigkeit offener massiver Bogenbriicken," Schweizerische Bau1.eitung, Vol. 77, No. 15 and 16, 1921.

31. PoweD, G., and Klingner,R., "Elastic Lateral Buckling of Steel Beams," Journal of the Structural Division, ASCE, Vol. 96, No. ST9, Proc. Paper 7555 1970 pp. 1919-1932. ' ,

32. Przemieniecki, J. S., Theory of Matrix Structural Analysis, McGraw-Hill Publishing Co., New York, N.Y., 1968.

33. Raj~karan, S., "Finite Element Analysis of Thin-Walled Members of Open Section," tlles!s presented to the University of Alberta, at Edmonton, Canada, in 1971, in partial fulf"illment of the requirements for the decree of Doctor of Philosophy.

34. Renton, J. D., "Bucklin& of Frames Composed of Thin-Walled Members," Thin-Walled Structures, A. H. Chilver, J. WJJey, ed., John WI1ey and Sons, Inc., New York, N.Y., 1967, pp. I-59 (see Eqs. 130-141).

35. Sabir, A. B., and AshweU, D. G., "A Comparison of Curved Beam Finite EJements when used in Vibration Problems," JOfU'ftlll of Sound and Vibration, Vol. IS, 1971, pp. 555-S63.

36. Stiissi,. F., Kippen lind Qwncllwingungen von Bogentriigern,International Association for Bridge and Structural Ensinurirtg PllbUcations, Vol. 7, 1943, pp. 327-343.

37. Walker, A. C., "A NonHnear Finite Element Analysis of Shallow Circular Arches" International JOfD'I'UJI of Solids and Structures, Vol. 5, 1969, pp. 97-107. '

38. Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, 2nd ed. McGraw-Hili Publishina Co., New York, N.Y., 1961, pp. 212-277. '

39. Urban, I. V., "Theory of the Analysis of Thin-Walled Bar Structw"es" (in Russian), Izd. TZDI, Moscow, U.S.S.R., 1955.

40. V1asov, V. Z., Thin-Walled Elastic Ban (in Russian), 2nd ed., Fizmatgiz, Moscow, U.S.S.R .. 1959 (also, English translation, Israel Prosram for Scientific Translations, Jerusalem, Israel, 1961; French translation. Eyrolles, Paris. France, 19(2).

41. Zienkiewicz, O. C., The Finite Element Method in Engineering Science, McGraw-Hili Publishing Co., London, England, 1971.

API'EMlIX •• --a..c NmATICM

The following symbols are u.sed in this paper:

A,E,G a, b, c

BO, At,! , w" po

B, M •. My, P C,C1

F GI,

I., I,

I, L

cross-sectional area, Young's and shear modulus; unit vectors in Eq. 24 (Fag. 6); initial birnoment, bending moments and axial force (Eq. 2), (Fig. I); incremental values of BO, At,! , w" PO; parameters given by Eq. 17; column matrix of generalized forces, Eq. 18; Saint-Venant's rigidity in simple torsion; principal cross-sectional moments of inertia (about centroidal axes x and y); I A cO 2 dA = principal sectorial moment of inertia (referred to shear center); EI., = warping rigidity of Timoshenko and Gere (38); element stiffness matrix and geometrical stiffness matrices (Eq. 14); length of element or length of beam along are, respectively;


M" M, q

R,R r =

S, T, T'

Subscripts.

t

Vx ' v y

W,W x, y,Z

x, y, Z

w,w

i, j, k n

( r)

Superscripts. o

()'

torque about shear center and St. Venant's torque; column.matrix of generalized displacements (Eq. 11); radius of curvature and rotation matrix, Eq. 24; parameter given by Eq. 9; . state vector, transformation and transfer matnces, Eqs. 20,27, 19; . matrix of transformation from shear center to centrOid, given by Eq. 25; lateral and longitudinal displacements, Eqs. 7. 1 (Fig.

1); shear forces in x and y directions;

parameters given by Eq. 9; global cartesian coordinates (Fig. 6); local cartesian coordinates of cross section or element (Fig. I); coordinates of shear center; parameters given by Eq. 10; incremental potential energy (Eq. 3);

parameter given by Eq. 17; . . displacements of cross section and Its rotatIon about axis z (Fig. I); and . sectorial coordinate (Eq. I) with respect to centrOId, and to shear center (w differs only by constant from warping function W J of Timoshenko and Gere (38),

their use of bars is different).

joints; elements; and load increments.

initial internal forces; quantities referred to shear center, without bar, referred to centroid; and i) / az.

--------------------------------------------------------------------------6----------; 10147 BUCKLING OF mIN·WALLED BEAMS AND FRAMES :

I

KEY WORDS: Arches; Beams (supports); BIlCkIiq; Curved beams; Elastic : properties; Eqiaeeria& aeduuaics; Finite elements; Frames; Matrb metJlods I

(struetaral); Stability; Thill-wall structures :

ABSTRACf: The potential energy expression and the (14 by 14) stiffness matrix of a straight thin-walled beam element of open asymmetric cross section, subjected to initial axial force, initial bending moments, and initial bimoment, are derived. The transformation matrix relating the forces and displacements (including bimoment and warping parameter) at the adjacent end cross section of two elements meeting at an angle is deduced as the limiting case of a transfer matrix of a curved beam. To cope with asymmetric cross sections, some element displacements and forces are referred to the shear center and others to the cross-sectional centroid, and the matrix for transformation from shear center to centroid is set up. The incremental largedisplacement analysis is formulated using the Eulerian coordinate approach with updating of the local coordinate systems at each load increment. The deformed beams are imagined to be composed of straight elements. Results of lateral post-buckling analysis of various beams are presented.

REFERENCE: Bazant. Zdenek P., and El Nimeiri, Mahjoub, "Large-Deflection Spatial B:JCilling of Thin Walled Beams and Frames," Journal of the Engineering Mech4l1ics Division, ASeE, Vol. 99, No. EM6, Proc. Paper 10147, December, 1973, pp. 1259-1281

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Documents

JOURNAL OF 'THE ENGINEERING MECHANICS · PDF fileJOURNAL OF 'THE ENGINEERING MECHANICS DIVISION LARGE-DEFLECTION SPATIAL BUCKLING OF ... studIes of large deflectIons and post-buckling